Jet Propulsion/1D Analysis

Ramjet

The thrust produced by a ramjet is

$T = \dot{m} u_{inlet} ( \sqrt{\tau_b} - 1 )$

where τb is the the total temperature ratio produced by the combustor. Metallurgy and the availability of cooling will limit the maximum temperature that can be sustained in the combustor. We can define τmax as the maximum total temperature ratio compared to the inlet conditions and τc as the temperature ratio relating the static and total temperatures. Then

$\tau_{max} = \tau_b \tau_c = \tau_b \left( 1 + \frac {\gamma-1}{2} M_0^2 \right)$

Thus as τc increases with speed for a fixed maximum temperature τmax stays constant and τb must reduce. If the theoretical frontal area of the ramjet is constant then the mass flow through the ramjet will increase linearly with the Mach number. At the same time the heat added diminishes. The thrust then is:

$T = A \rho (M_0 a_0)^2 \left[ \sqrt{\frac{\tau_{max}}{1 + \frac {\gamma-1}{2} M_0^2}} - 1 \right]$

When τmaxc the term in brackets goes to zero and the thrust vanishes. The thrust for a given τmax is shown in the following figure (the dashed line is the peak thrust):